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Comment by _blz2 | Hacker News Reader

For some reason this seems incredibly funny to me. I can imagine mathematicians laughing at this too.

Computer science is more rigorous than mathematics. The proofs written for computers have to actually work in computer, which is a collection of mathematics that have provably been demostrated to retain their fundamental basic properties that allow computation to both define and test the validity of statements.

If so, then that's because traditional mathematical proofs are arguments meant to convince humans of the validity of a claim. They are formal only insofar as e.g. law is formal.

People's minds have a limit to how much information they can retain. Pushing new information in comes at the expense of pushing other information out. Computer science isn't limited this way because every simplified property of mathematics is retained mechanically. If a property is created that is invalid, computation ceases to be computation. Minds on the other hand, aren't so limited, but that doesn't mean that what they produce is rigorous (or correct, with respect to the context they've been defined in).

I agree with you, if that's not clear. I don't understand your law analogy precisely, but I'm assuming in a general sense you mean 'human defines rule set, rule set retains it's properties provided that humans continue to enforce rule set'. To my mind, that is so open to variation, my brain may begin to attack itself. Humans can 'make sense' of anything. Computation needs to be able to do what it does regardless of whether a human is there to provide input and verify it 'makes sense' to the mind that defined it.

Universe turning into paperclips, sure, but the fact that a set of properties can be used to recreate themselves through their own definitions, that's beautiful, that's computation, rigor. These are things I used to love about math, but computer science just does math better than math does.

Math is built on a far more rigorous logical foundation than "this program ran on a physical computer, which proves that the statement is valid". Every valid mathematical theorem is a consequence of, and can be traced back to, a set of definitions and well-defined logical axioms. The parts of computer science that are most rigorous are strict subsets of math, e.g. computability theory, logic programming, formal languages, etc.

Most of those subsets of math are relatively new and grew alongside or developed into computation (in fairly quick succession, in comparison to the history of mathematics). In addition, those fields are still actively being worked on as there are still problems that can not be resolved with mathematics alone, and much of that rests on computer science being able to retain much of the structure of mathematics without reinforcing foundations that are problematic or open problems. The mind that sees those problems in computation is a mind that is looking for problems because it is comparing computation to knowledge of mathematics, when there is actually a union between the two that can be provably demonstrated to be built from a foundation that is purely calculable / computable.

Show me the foundations for the continuum hypothesis.

What you're arguing for is a view of mathematics that has been dead for a century now. With the Godels incompleteness theorem and Turing halting problem show you that there are cases of 'true' statements in mathematics that can't be reduced to "well-defined logical axioms".

The continuum hypothesis is a well-defined logical axiom, occasionally.

My argument is about the rigor of theorems. ZFC may be incomplete, but every existing theorem that builds up the entirety of modern mathematics is (in theory, admittedly) built upon a rigorous chain of truth, traced back to those axioms and definitions.

Most mathematicians who don't study metamathematics or philosophy of math have no reason to ever think about formalism or computability.

An aside for people who think Godel's incompleteness theorem somehow invalidates math: don't forget about the much lesser known Godel's completeness theorem, which states that if something is true in every model

Most mathematicians who don't study metamathematics or philosophy of math have no reason to ever think about formalism or computability. Godel's completeness theorem stipulates that there are self-contained consistent systems within the language wherein all of their nice proofs are true and universal, and where plenty of truth still needs to be discovered.

Re: Continuum hypothesis: the way I try to explain this is, take the statement S = "x^2 = -1 has no solutions". In Z, S is true. In C, S is false. What does it mean to say "S is true" without specifying the system? It depends on which underlying axioms you choose. There are extensions of ZFC where you can prove CH is true, and extensions where you can prove CH is false. In both extensions, the proofs are rigorous and valid, and there is plenty of valid mathematics to be done.

But ok, my point is, how can you say the CH is a true statement that can't be reduced to axioms? What does it mean to be "true" besides that something follows from other truths? The choice of axioms matters, sure, but everything that's true within a system follows from the axioms of that system.

There are two ways to see it. But that's also the problem.

> What does it mean to be "true" besides that something follows from other truths?

I agree with you, but I also don't. You know as well as I, that 'true' can mean something besides 'that which follows from other truths'. All the rules you have to rely on (such as'implication) - that's a truth you are dependent on for math to work, but can not define within math. Implication is a fundamental foundation. But can you define implication without using the concept of implication?

> The choice of axioms matters, sure, but everything that's true within a system follows from the axioms of that system.

It's easy to point out flaws in reasoning. It's so, so much harder to have an airtight reasoning system, that goes for mathematics and computation, both, together, alone, etc.

> What does it mean to be "true" besides that something follows from other truths?

Truth is true, no more, no less. Once you turn it into symbols, it turns into a mess (or a work of art).

>Computer science is more rigorous than mathematics

Not true. Also computer science is very vague in a way.

To you.

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